Methods of enumeration of spanning trees in a finite graph and relations to various areas of mathematics and physics have been investigated for more than 150 years. We will review the history and applications. Then we will give new formulas for the asymptotics of the number of spanning trees of a graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call “tree entropy', which we show is a logarithm of a normalized determinant of the graph Laplacian for infinite graphs. Proofs involve new traces and the theory of random walks.